Counting rank two local systems with at most one, unipotent, monodromy

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Abstract

The number of rank two ℚl-local systems, or ℚl-smooth sheaves, on (X — {u}) ⊗𝔽q 𝔽, where X is a smooth projective absolutely irreducible curve over 𝔽q, 𝔽 an algebraic closure of 𝔽q and u is a closed point of X, with principal unipotent monodromy at u, and fixed by Gal(𝔽/𝔽q), is computed. It is expressed as the trace of the Frobenius on the virtual ℚl-smooth sheaf found in the author’s work with Deligne on the moduli stack of curves with etale divisors of degree M ≥ 1. This completes the work with Deligne in rank two. This number is the same as that of representations of the fundamental group π1((X — {u}) ⊗𝔽q 𝔽) invariant under the Frobenius Frq with principal unipotent monodromy at u, or cuspidal representations of GL(2) over the function field 𝔽 = 𝔽q(X) of X over 𝔽q with Steinberg component twisted by an unramified character at u and unramified elsewhere, trivial at the fixed idèle α of degree 1. This number is computed in Theorem 4.1 using the trace formula evaluated at fuΠv≠uχKv, with an Iwahori component fu = χIu/\Iu\, hence also the pseudo-coefficient χIu/\Iu\ — 2 χKu of the Steinberg representation twisted by any unramified character, at u. Theorem 2.1 records the trace formula for GL(2) over the function field F. The proof of the trace formula of Theorem 2.1 recently appeared elsewhere. Theorem 3.1 computes, following Drinfeld, the number of ℚl-local systems, or ℚl-smooth sheaves, on X ⊗Fq 𝔽, fixed by Frq, namely ℚl-representations of the absolute fundamental group π(X ⊗𝔽q 𝔽) invariant under the Frobenius, by counting the nowhere ramified cuspidal representations of GL(2) trivial at a fixed idele α of degree 1. This number is expressed as the trace of the Frobenius of a virtual ℚl-smooth sheaf on a moduli stack. This number is obtained on evaluating the trace formula at the characteristic function Πv χKV of the maximal compact subgroup, with volume normalized by |Kv| = 1. Section 5, based on a letter of P. Deligne to the author dated August 8, 2012, computes the number of such objects with any unipotent monodromy, principal or trivial, in our rank two case. Surprisingly, this number depends only on X and deg(S), and not on the degrees of the points in S1.

Original languageEnglish
Pages (from-to)739-763
Number of pages25
JournalAmerican Journal of Mathematics
Volume137
Issue number3
DOIs
StatePublished - 1 Jan 2015

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